Archive for March, 2009

As I have already commented, the outcome of the Polymath experiment differed in one important respect from what I had envisaged: though it was larger than most mathematical collaborations, it could not really be described as massive. However, I haven’t given up all hope of a much larger collaboration, and in this post I want to think about ways that that might be achieved.

First, let me say what I think is the main rather general reason for the failure of Polymath1 to be genuinely massive. I had hoped that it would be possible for many people to make small contributions, but what I had not properly thought through was the fact that even to make a small contribution one must understand the big picture. Or so it seems: that is a question I would like to consider here. (more…)

Very many thanks to all who have written in pointing out errors in the Princeton Companion. I am told that it won’t be too long before the next printing, so I have finally been forced to collect together the errata in a systematic way. I am copying the list I have just sent to Princeton University Press so that if anyone finds an error now they can easily check whether it has already been spotted. (I was asked if I would do this some time ago — now at last I have.) The list appears after the break. From now on if people point out further errors I will add them to the list, with some indication of whether they have yet been corrected.

The secondary purpose of this post is to suggest that you should wait a bit if you are thinking of buying the book. I don’t want to hit sales too hard, but I’m guessing that not all PCM buyers are avid readers of this blog so I might as well reward those who are. Of course, you could take the attitude that the error-strewn version is a collector’s item: if so, hurry while stocks last. (more…)

I’m not sure how many more comment threads we will have, but we are running out of the 1000-1049 thread, so it’s time for a new one. The main news to report since the last post is that progress is being made on writing up the proof of DHJ(3) and DHJ(k). At the moment it is more like preparatory sketches, but they are pretty detailed and can, as usual, be found on the wiki. It looks as though some of the more technical parts will end up very streamlined thanks to work of Ryan O’Donnell: the final proof of DHJ(k) should be surprisingly short (though I hope that we will write it up with plenty of accompanying explanation so that it is not too compressed and hard to understand).

In this post I want to discuss some general issues that arise naturally in the light of how the polymath experiment has gone so far. First, let me say that for me personally this has been one of the most exciting six weeks of my mathematical life. That is partly because it is always exciting to solve a problem, but a much more important reason is the way this problem was solved, with people chipping in with their thoughts, provoking other people to have other thoughts (sometimes almost accidentally, and sometimes more logically), and ideas gradually emerging as a result. Incidentally, many of these ideas are still to be properly explored: at some point the main collaboration will probably be declared to be over (though I suppose in theory it could just go on and on, since its seems almost impossible to clear up every interesting question that emerges) and then I hope that the comments will be a useful resource for anybody who wants to find some interesting open problems. (more…)

Without anyone being particularly aware of it, a race has been taking place. Which would happen first: the comment count reaching 1000, or the discovery of a new proof of the density Hales-Jewett theorem for lines of length 3? Very satisfyingly, it appears that DHJ(3) has won. If this were a conventional way of producing mathematics, then it would be premature to make such an announcement — one would wait until the proof was completely written up with every single i dotted and every t crossed — but this is blog maths and we’re free to make up conventions as we go along. So I hereby state that I am basically sure that the problem is solved (though not in the way originally envisaged).

Why do I feel so confident that what we have now is right, especially given that another attempt that seemed quite convincing ended up collapsing? Partly because it’s got what you want from a correct proof: not just some calculations that magically manage not to go wrong, but higher-level explanations backed up by fairly easy calculations, a new understanding of other situations where closely analogous arguments definitely work, and so on. And it seems that all the participants share the feeling that the argument is “robust” in the right way. And another pretty persuasive piece of evidence is that Tim Austin has used some of the ideas to produce a new and simpler proof of the recurrence result of Furstenberg and Katznelson from which they deduced DHJ. His preprint is available on the arXiv.

Better still, it looks very much as though the argument here will generalize straightforwardly to give the full density Hales-Jewett theorem. We are actively working on this and I expect it to be done within a week or so. (Work in progress can be found on the polymath1 wiki.) Better even than that, it seems that the resulting proof will be the simplest known proof of Szemerédi’s theorem. (There is one other proof, via hypergraphs, that could be another candidate for that, but it’s slightly less elementary.)

I have lots of thoughts about the project as a whole, but I want to save those for a different and less mathematical post. This one is intended to be the continuation of the discussion of DHJ(3), and now DHJ(k), into the 1000s. More precisely, it is for comments 1000-1049.

Once again there is not a huge amount to say in this post. Since the last post there have been a few additions to the polymath1 wiki that may be of some use. In particular, there is now a collection of fairly complete write-ups of related results (see the section entitled “Complete proofs or detailed sketches of potentially useful results”) to which I hope we will add soon. Also on the wiki is an account of the Ajtai-Szemerédi proof of the corners theorem, which seems to have some chance of serving as a better model for a proof of DHJ(3) than the proof via the triangle-removal lemma. Meanwhile, progress has been made in understanding and to some extent combinatorializing the ergodic-theoretic proof of DHJ(3), ideas from which have fed into the discussion. As with the last post, this one is mainly to stop the number of comments getting too large. We’re now down to 50 comments per post (except that it was 51 for the last one and will be 49 for this), since with the new threading we seem to be averaging at least one reply per comment.